Question

If a heat engine and a refrigerator are working between the same two temperatures \(T_1\) and \(T_2\) (\(T_1>T_2\)), then the ratio of efficiency of heat engine to coefficient of performance of refrigerator is

• A. \(\frac{(T_1 - T_2)}{T_1 T_2} \)
• B. \(\frac{(T_1 + T_2)}{T_1 T_2} \)
• C. \(\frac{(T_1 - T_2)^2}{T_1 T_2} \)
• D. \(\frac{(T_1 + T_2)^2}{T_1 T_2} \)

Hint:

Remember the formulas for the efficiency of a Carnot heat engine and the coefficient of performance of a Carnot refrigerator/heat pump. Heat Engine Efficiency ($\eta$): \(\eta = 1 - \frac{T_{cold}}{T_{hot}}\) Refrigerator COP ($\beta$): \(\beta = \frac{T_{cold}}{T_{hot} - T_{cold}}\) Heat Pump COP ($COP}_{HP}$): \(COP}_{HP} = \frac{T_{hot}}{T_{hot} - T_{cold}}\) Also, note the relationship: \(COP}_{HP} = \beta + 1\) and \(\eta = \frac{1}{COP}_{HP}}\) is incorrect. The correct relationship between efficiency and COP of a refrigerator is \(\eta = \frac{1}{1 + COP}_{ref}}\).

Answer 1

The Correct Option is C

Step 1: Define the efficiency of a heat engine.
For a reversible heat engine (like a Carnot engine) operating between a hot reservoir at temperature \(T_1\) and a cold reservoir at temperature \(T_2\), where \(T_1>T_2\), the efficiency \(\eta\) is given by: \[ \eta = 1 - \frac{T_2}{T_1} = \frac{T_1 - T_2}{T_1} \] Step 2: Define the coefficient of performance (COP) of a refrigerator.
For a reversible refrigerator operating between the same two temperatures \(T_1\) and \(T_2\), the coefficient of performance (\(\beta\) or COP) is given by: \[ \beta = \frac{T_2}{T_1 - T_2} \] Step 3: Calculate the ratio of the efficiency of the heat engine to the coefficient of performance of the refrigerator.
We need to find the ratio \(\frac{\eta}{\beta}\).
Substitute the expressions for \(\eta\) and \(\beta\) from Step 1 and Step 2:
\[ \frac{\eta}{\beta} = \frac{\frac{T_1 - T_2}{T_1}}{\frac{T_2}{T_1 - T_2}} \] To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator: \[ \frac{\eta}{\beta} = \frac{(T_1 - T_2)}{T_1} \times \frac{(T_1 - T_2)}{T_2} \] Multiply the terms: \[ \frac{\eta}{\beta} = \frac{(T_1 - T_2)^2}{T_1 T_2} \] The final answer is $\boxed{\frac{(T_1 - T_2)^2}{T_1 T_2}}$.